Split-Compensated Controlled Partial b-Metric Spaces and Related Fixed Point Results

Résumé

Inspired by the generalized contraction framework on partial metric spaces developed
by Altun, Sola, and Simsek, we introduce a new generalized distance structure that we call
a split-compensated controlled partial b-metric space. The new setting modifies the usual
compensated triangular inequality by allowing the self-distance correction to be distributed
between the intermediate point and the endpoints. After presenting the basic definition and
illustrative examples, we establish a Banach-type fixed point theorem on 0-complete spaces.
We then derive two nonlinear fixed point principles of Boyd–Wong/Matkowski type and
Hardy–Rogers maximum type. In addition, we prove convergence of the Picard iteration,
uniqueness of the fixed point, and a well-posedness estimate for approximate fixed points.
Several considered examples are included, together with applications to nonlinear Volterra
and Hammerstein integral equations on cones of nonnegative continuous functions. The paper
is intended as a self-contained candidate framework for further investigation of nonlinear
contractions in generalized partial metric geometry.